Harmonic Oscillators¶

In this tutorial, we demo how to generate data of harmonic oscillators.

In [1]:

import pandas as pd
import plotly.express as px

from hamilflow.models.harmonic_oscillator import (
    DampedHarmonicOscillator,
    SimpleHarmonicOscillator,
)

import pandas as pd import plotly.express as px from hamilflow.models.harmonic_oscillator import ( DampedHarmonicOscillator, SimpleHarmonicOscillator, )

/tmp/ipykernel_2811/2648449548.py:1: DeprecationWarning: 
Pyarrow will become a required dependency of pandas in the next major release of pandas (pandas 3.0),
(to allow more performant data types, such as the Arrow string type, and better interoperability with other libraries)
but was not found to be installed on your system.
If this would cause problems for you,
please provide us feedback at https://github.com/pandas-dev/pandas/issues/54466
        
  import pandas as pd

In [2]:

n_periods = 3
n_samples_per_period = 200

n_periods = 3 n_samples_per_period = 200

Simple Harmonic Oscillator¶

For an simple harmonic oscillator, the action of a simple harmonic oscillator is

$$S_L[x] = \int_{t_0}^{t_1} \mathbb{d}t \left\{\frac{1}{2} m \dot x^2 - \frac{1}{2} m \omega^2 x^2 \right\}\,,$$

where the least action principle leads to the following equation of motion,

$$ \ddot x + \omega^2 x = 0\,. $$

A simple harmonic oscillator is a periodic motion.

In [3]:

sho_omega = 0.5

sho = SimpleHarmonicOscillator(system={"omega": sho_omega})

sho_omega = 0.5 sho = SimpleHarmonicOscillator(system={"omega": sho_omega})

In [4]:

df_sho = sho(n_periods=n_periods, n_samples_per_period=n_samples_per_period)
df_sho.head()

df_sho = sho(n_periods=n_periods, n_samples_per_period=n_samples_per_period) df_sho.head()

Out[4]:

	t	x
0	0.000000	1.000000
1	0.062832	0.999507
2	0.125664	0.998027
3	0.188496	0.995562
4	0.251327	0.992115

In [5]:

px.line(
    df_sho,
    x="t",
    y="x",
    title=rf"Simple Harmonic Oscillator (omega = {sho_omega})",
    labels={
        "x": r"Displacement $x(t)$",
        "t": r"$t$",
    },
)

px.line( df_sho, x="t", y="x", title=rf"Simple Harmonic Oscillator (omega = {sho_omega})", labels={ "x": r"Displacement $x(t)$", "t": r"$t$", }, )

Damped Harmonic Oscillator¶

A damped harmonic oscillator is a simple harmonic oscillator with damping force that is proportional to its velocity,

$$ \ddot x + \omega^2 x = - 2\xi\omega \dot x\,. $$

In this section, we demonstrate three scenarios of a damped harmonic oscillator.

In [6]:

dho_systems = {
    "Underdamped": {"omega": 0.5, "zeta": 0.2},
    "Critical Damped": {"omega": 0.5, "zeta": 1},
    "Overdamped": {
        "omega": 0.5,
        "zeta": 1.2,
    },
}

dfs_dho = []

for s_name, s in dho_systems.items():

    dfs_dho.append(
        DampedHarmonicOscillator(system=s)(
            n_periods=n_periods, n_samples_per_period=n_samples_per_period
        ).assign(system=rf"{s_name} (omega = {s.get('omega')}, zeta = {s.get('zeta')})")
    )

fig = px.line(
    pd.concat(dfs_dho),
    x="t",
    y="x",
    color="system",
    title=rf"Damped Harmonic Oscillator",
    labels={
        "x": r"Displacement $x(t)$",
        "t": r"$t$",
    },
)
fig.update_layout(legend={"yanchor": "top", "y": -0.2, "xanchor": "left", "x": 0})

dho_systems = { "Underdamped": {"omega": 0.5, "zeta": 0.2}, "Critical Damped": {"omega": 0.5, "zeta": 1}, "Overdamped": { "omega": 0.5, "zeta": 1.2, }, } dfs_dho = [] for s_name, s in dho_systems.items(): dfs_dho.append( DampedHarmonicOscillator(system=s)( n_periods=n_periods, n_samples_per_period=n_samples_per_period ).assign(system=rf"{s_name} (omega = {s.get('omega')}, zeta = {s.get('zeta')})") ) fig = px.line( pd.concat(dfs_dho), x="t", y="x", color="system", title=rf"Damped Harmonic Oscillator", labels={ "x": r"Displacement $x(t)$", "t": r"$t$", }, ) fig.update_layout(legend={"yanchor": "top", "y": -0.2, "xanchor": "left", "x": 0})

In [ ]: